The Number of Solutions to the Trinomial Thue Equation

TL;DR

This paper investigates the solutions to a specific class of trinomial Thue equations, providing improved explicit upper bounds on the number of integer solutions for forms of degree at least 3, especially for large degrees.

Contribution

It refines existing bounds on the number of solutions to trinomial Thue equations, reducing the maximum solutions count for degrees n ≥ 219, advancing prior results by Emery Thomas.

Findings

No more than 32 solutions for odd n ≥ 219

No more than 40 solutions for even n ≥ 219

Improves previous bounds of 38 and 48 solutions respectively

Abstract

In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible (over $\mathbb{Z}$) binary form with degree $n \geqslant 3$ and exactly three nonzero summands. In particular, we improve Emery Thomas' explicit upper bounds on the number of solutions to this equation. For instance, when $n \geqslant 219$, we show that there are no more than 32 integer pair solutions to this equation when $n$ is odd and no more than 40 integer pair solutions to this equation when $n$ is even, an improvement on Thomas' work, where he shows that there are no more than 38 such solutions when $n$ is odd and no more than 48 such solutions when $n$ is even.